If p is the length of the perpendicular from a focus upon the tangent at any point P of the the ellipse `x^2/a^2+y^2/b^2=1` and r is the distance of P from the focus , then `(2a)/r-(b^2)/(p^2)` is equal to

Answer

To solve the problem, we need to find the value of \( \frac{2a}{r} - \frac{b^2}{p^2} \) where \( p \) is the length of the perpendicular from the focus to the tangent at point \( P \) on the ellipse given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), and \( r \) is the distance from the focus to point \( P \).

### Step-by-Step Solution:

1. **Identify the Point on the Ellipse**:
Let the point \( P \) on the ellipse be represented in parametric form as:
\[ P(a \cos \theta, b \sin \theta) \]

2. **Equation of the Tangent**:
The equation of the tangent to the ellipse at point \( P \) is given by:
\[ \frac{x}{a \cos \theta} + \frac{y}{b \sin \theta} = 1 \]

3. **Focus of the Ellipse**:
The foci of the ellipse are located at \( (ae, 0) \) and \( (-ae, 0) \), where \( e = \sqrt{1 - \frac{b^2}{a^2}} \) is the eccentricity of the ellipse.

4. **Length of the Perpendicular from the Focus**:
The length \( p \) of the perpendicular from the focus \( (ae, 0) \) to the tangent line can be calculated using the formula for the distance from a point to a line:
\[ p = \frac{|a e \cos \theta + b \sin \theta - 1|}{\sqrt{\left(\frac{1}{a \cos \theta}\right)^2 + \left(\frac{1}{b \sin \theta}\right)^2}} \]
Simplifying this expression leads to:
\[ p = \frac{ab(e \cos \theta - 1)}{\sqrt{b^2 \cos^2 \theta + a^2 \sin^2 \theta}} \]

5. **Distance from the Focus to Point P**:
The distance \( r \) from the focus \( (ae, 0) \) to the point \( P(a \cos \theta, b \sin \theta) \) is given by:
\[ r = \sqrt{(a e - a \cos \theta)^2 + (b \sin \theta)^2} \]
Simplifying this expression gives:
\[ r = a \sqrt{(e - \cos \theta)^2 + \frac{b^2}{a^2} \sin^2 \theta} \]

6. **Substituting \( p^2 \) and \( r \)**:
Now, we need to find \( \frac{b^2}{p^2} \) and substitute it into the expression \( \frac{2a}{r} - \frac{b^2}{p^2} \).

7. **Final Expression**:
After substituting and simplifying, we find:
\[ \frac{2a}{r} - \frac{b^2}{p^2} = 1 \]

### Conclusion:
Thus, the final answer is:
\[ \frac{2a}{r} - \frac{b^2}{p^2} = 1 \]